Alice is standing on the circumference of a large circular room of radius $10$. There is a circular pillar in the center of the room of radius $5$ that blocks Alice’s view. The total area in the room Alice can see can be expressed in the form $\frac{m\pi}{n} +p\sqrt{q}$, where $m$ and $n$ are relatively prime positive integers and $p$ and $q$ are integers such that $q$ is square-free. Compute $m + n + p + q$. (Note that the pillar is not included in the total area of the room.) [img]https://cdn.artofproblemsolving.com/attachments/5/1/26e8aa6d12d9dd85bd5b284b6176870c7d11b1.png[/img]

A path from $ (0,0)$ to $ (n,n)$ on the lattice is made up of unit moves upward or rightward. It is balanced if the sum of the x-coordinates of its $ 2n\plus{}1$ vertices equals the sum of their y-coordinates. Show that a balanced path divides the square with vertices $ (0,0)$, $ (n,0)$, $ (n,n)$, $ (0,n)$ into two parts with equal area.

A positive integer is called [i]downhill[/i] if the digits in its decimal representation form a nonstrictly decreasing sequence from left to right. Suppose that a polynomial $P(x)$ with rational coefficients takes on an integer value for each downhill positive integer $x$. Is it necessarily true that $P(x)$ takes on an integer value for each integer $x$?

[b]1.[/b] Let $a$ ( $\neq e$, the unit element) be an element of finite order of a group $G$ and let $t$ ($\geq 2$) be a positive integer. Show: if the complex $A= \{ e,a,a^2, \dots , a^{t-1} \} $ is not a group, then for every positive integer $k$( $2 \leq k \leq t$) the complex $B= \{ e. a^k, a^{2k}, \dots , a^{(t-1)k} \} $ differs from $A$. [b](A. 16)[/b]

Square $ABCD$ has side length $5$. Draw E on $BC$ and $F$ on $AD$ such that $BE < AF$. Next, flip $ABCD$ across $EF$ to a square $A'B'C'D'$ such that $C'$ lies in the interior of $ABCD$ and $C$ lies in the interior of $A'B'C'D'$. Suppose that $CC' = 4$ and $DD' = 2$. Compute $AA'$.

Find the natural numbers $x, y, z$ that verify the equation: $$2^x + 3 \cdot 11^y =7^z$$

Let $\triangle ABC$ be an acute triangle. Point $D$ is arbitrarily chosen on the side $BC$. Let the circumcircle of the triangle $\triangle ADB$ intersect the segment $AC$ at $M$, and the circumcircle of the triangle $\triangle ADC$ intersect the segment $AB$ at $N$. Prove that the tangents of the circumcircle of the triangle $\triangle AMN$ at $M$ and $N$ intersect at a point that lies on the line $BC$.

A circle with center $O$ is tangent to the sides $AB$, $BC$, $AC$ of a triangle $ABC$ at points $E,F,D$ respectively. The lines $AO$ and $CO$ meet $EF$ at points $N$ and $M$. Prove that the circumcircle of triangle $OMN$ and points $O$ and $D$ lie on a line.

What is the largest integer with distinct digits such that no two of its digits sum to a perfect square?

Let $ABC$ be a triangle with $BC>CA>AB$ and let $G$ be the centroid of the triangle. Prove that \[ \angle GCA+\angle GBC<\angle BAC<\angle GAC+\angle GBA . \] [i]Dinu Serbanescu[/i]

Let $r_1$, $r_2$, $r_3$ be the distinct real roots of $x^3-2019x^2-2020x+2021=0$. Prove that $r_1^3+r_2^3+r_3^3$ is an integer multiple of $3$.

Joshua likes to play with numbers and patterns. Joshua's favorite number is $6$ because it is the units digit of his birth year, $1996$. Part of the reason Joshua likes the number $6$ so much is that the powers of $6$ all have the same units digit as they grow from $6^1$: \begin{align*}6^1&=6,\\6^2&=36,\\6^3&=216,\\6^4&=1296,\\6^5&=7776,\\6^6&=46656,\\\vdots\end{align*} However, not all units digits remain constant when exponentiated in this way. One day Joshua asks Michael if there are simple patterns for the units digits when each one-digit integer is exponentiated in the manner above. Michael responds, "You tell me!" Joshua gives a disappointed look, but then Michael suggests that Joshua play around with some numbers and see what he can discover. "See if you can find the units digit of $2008^{2008}$," Michael challenges. After a little while, Joshua finds an answer which Michael confirms is correct. What is Joshua's correct answer (the units digit of $2008^{2008}$)?

Find the smallest value that the expression takes $x^4 + y^4 - x^2y - xy^2$, for positive numbers $x$ and $y$ satisfying $x + y \le 1$.

Claire takes a square piece of paper and folds it in half four times without unfolding, making an isosceles right triangle each time. After unfolding the paper to form a square again, the creases on the paper would look like

Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that \[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \] Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.

There are $19$ balls in a box, numbered $1$ through $19$. When we go out get that box without looking five different balls, which number has the largest probability of being the difference between the highest and lowest number drawn? Justify you reply .

Let $\theta$ and $p(p<1)$ ) be nonnegative real numbers. Suppose that $f:X\to Y$ is mapping with $f(0)=0$ and $$\left |\left| 2f\left (\frac{x+y}{2}\right )-f(x)-f(y) \right |\right|_Y \leq \theta\left (\left |\left |x\right |\right |_X^p +\left |\left |y\right |\right |_X^p \right )$$ for all $x$, $y\in \mathbb{Z}$ with $x\perp y$ where $X$ is an orthogonality space and $Y$ is a real Banach space. Prove that there exists a unique orthogonally Jensen additive mapping $T:X\to Y$, namely a mapping $T$ that satisfies the so-called orthogonally Jensen additive functional equation $$2f\left (\frac{x+y}{2}\right )=f(x)+f(y)$$for all $x$, $y\in \mathbb{X}$ with $x\perp y$, satisfying the property $$\left |\left|f(x)-T(x) \right |\right|_Y \leq \frac{2^p\theta}{2-2^p}\left |\left |x\right |\right |_X^p$$ for all $x\in X$

For $f(x)=\frac{1}{x}\ (x>0)$, prove the following inequality. \[f\left(t+\frac 12 \right)\leq \int_t^{t+1} f(x)\ dx\leq \frac 16\left\{f(t)+4f\left(t+\frac 12\right)+f(t+1)\right\}\]

While Peter was driving from home to work, he noticed that after driving 21 miles, the distance he had left to drive was 30 percent of the total distance from home to work. How many miles was his complete trip home to work?

Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer.

Determine the number of all positive integers which cannot be written in the form $80k + 3m$, where $k,m \in N = \{0,1,2,...,\}$

There is secret society with $2011$ members. Every member has bank account with integer balance ( can be negative). Sometimes some member give one dollar to every his friend. It is known, that after some such moves members can redistribute their money arbitrarily. Prove, that there are exactly $2010$ pairs of friends.

Let $S$ be the set of positive integers $x$ for which there exist positive integers $y$ and $m$ such that $y^2-2^m=x^2$. (a) Find all of the elements of $S$. (b) Find all $x$ such that both $x$ and $x+1$ are in $S$.

A sequence $\{a_n\}$ is defined by: $a_1 = 1, a_{n+1} = a_n + \dfrac{1}{\sqrt{a_n}}$ for $n = 1, 2, 3, \ldots$. Find all real numbers $q$ such that the sequence $\{u_n\}$ defined by $u_n = a_n^q$, $n = 1, 2, 3, \ldots$ has nonzero finite limit when $n$ goes to infinity. THERE MIGHT BE A TYPO!

How many sequences of integers $1 \le a_1 \le a_2\le ... \le a_{11 }\le 2015$ that satisfy $a_i \equiv i^2$ (mod $12$) for all $1 \le i \le 11$ are there? (Le Anh Vinh)